Consider a drawing of graph \(G\) . Subdivide each edge into
segments, so that each segment is crossed at most once and each
subdivision point is incident to at most one crossed segment.
The so obtained drawing is IC-planar. Each edge of \(G\)  is
crossed at most by all other edges, since \(G\)  is simple. Thus \(O(n^2)\)  subdivisions per edge
suffice. For the lower bound consider the crossing number of
complete graphs \(K_n\) , which is \(\Omega (n^4)\)  [1]}, [2]}. Clearly, the number of crossings is unchanged by
subdivisions. Since \(k\) -planar and \(k\) -gap-planar graphs admit
\(O(m)\)  crossings, where \(m\)  is the number of edges, some edges have
\(\Omega (n^2)\)  crossings and need \(\Omega (n^2)\)  many subdivisions
for \(k\) -(gap)-planar graphs. Clearly, \(O(n^2)\)  subdivisions
suffice for fan-crossing graphs. For the lower bound consider
\(d\) -dimensional cube-connected cycles. By the lower bound on the
crossing number [3]}, there are edges with at least
\(\frac{1}{30} \frac{2^d}{d}-4\)  many crossings, which is \(c\frac{n}{\log ^2 n}\)  for some \(c>0\) , since \(n=d2^d\) . By Lemma
REF , 3-regular fan-crossing graphs are 2-planar,
so that at least \(cn / \log ^2 n\)  many subdivisions are needed for
some edges.
